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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge
Cambridge University Press
2016
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| Series: | Cambridge studies in advanced mathematics
157 |
| Subjects: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029226512&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Physical Description: | xiv, 471 Seiten Diagramme |
| ISBN: | 9781107019669 |
Staff View
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Record in the Search Index
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| adam_text | Titel: The three-dimensional Navier-Stokes equations
Autor: Robinson, James C
Jahr: 2016
The Three-Dimensional Navier-Stokes Equations Classical Theory JAMES C.^LOBINSON University of Warwick JOSÉ L. RODRIGO University of Warwick WITOLD SADOWSKI University of Warsaw Cambridge UNIVERSITY PRESS
Contents Preface page xiii Introduction 1 PART I WEAK AND STRONG SOLUTIONS Overview of Part I 17 1 Function spaces 19 1.1 Domain of the flow 19 1.2 Derivatives 20 1.3 Spaces of continuous and differentiable functions 21 1.4 Lebesgue spaces 23 1.5 Fourier expansions 26 1.6 Sobolev spaces W k,p 28 1.7 Sobolev spaces H s with s 0 29 1.8 Dual spaces 37 1.9 Bochner spaces 38 Notes 44 Exercises 44 2 The Helmholtz-Weyl decomposition 47 2.1 The Helmholtz-Weyl decomposition on the torus 48 2.2 The Helmholtz-Weyl decomposition in £2 c R 3 52 2.3 The Stokes operator 57 2.4 The Helmholtz-Weyl decomposition of L ? 63 Notes 66 Exercises 68 vii
70 70 73 79 82 84 86 87 89 89 92 98 101 103 107 108 111 112 115 117 122 124 125 125 127 128 133 137 140 143 144 145 146 Contents Weak formulation 3.1 Weak formulation 3.2 Basic properties of weak solutions 3.3 Alternative spaces of test functions 3.4 Equivalent weak formulation 3.5 Uniqueness of weak solutions in dimension two Notes Exercises Existence of weak solutions 4.1 The Galerkin method 4.2 Existence of weak solutions on bounded domains 4.3 The strong energy inequality 4.4 Existence of weak solutions on the whole space 4.5 The Aubin-Lions Lemma Notes Exercises The pressure 5.1 Solving for the pressure on T 3 and R 3 5.2 Distributional solutions in the absence of boundaries 5.3 Additional estimates on weak solutions 5.4 Pressure in a bounded domain 5.5 Applications of pressure estimates Notes Exercises Existence of strong solutions 6.1 General properties of strong solutions 6.2 Local existence of strong solutions 6.3 Weak-strong uniqueness and blowup 6.4 Global existence for small data in V 6.5 Global strong solutions in the two-dimensional case 6.6 Strong solutions on the whole space Notes Exercises
Contents ix 7 Regularity of strong solutions 148 7.1 Regularity in space 149 7.2 Regularity in space-time 153 Notes 155 Exercises 156 8 Epochs of regularity and Serrin’s condition 158 8.1 The putative set of singular times 158 8.2 The box-counting and Hausdorff dimensions 162 8.3 Epochs of regularity 166 8.4 More bounds on weak solutions 169 8.5 Serrin’s condition 170 8.6 Epochs of regularity on the whole space 175 Notes 176 Exercises 178 9 Robustness of regularity and convergence of Galerkin approximations 180 9.1 Robustness of strong solutions 180 9.2 Convergence of Galerkin approximations 184 Notes 188 Exercises 190 10 Local existence and uniqueness in H x/2 192 10.1 Critical spaces 192 10.2 Fractional Sobolev spaces and criticality of H ]/2 194 10.3 Local existence for initial data in H x/1 195 10.4 An auxiliary ODE lemma 201 Notes 202 Exercises 204 11 Local existence and uniqueness in L 3 206 11.1 Preliminaries 207 11.2 Local existence in 1? 208 11.3 A proof of Lemma 11.2 on T 3 215 Notes 216 Exercises 217
221 224 224 226 228 230 231 236 237 238 238 243 244 252 259 261 262 263 264 265 268 272 273 277 278 279 283 284 291 301 304 312 314 Contents PARTE LOCAL AND PARTIAL REGULARITY Overview of Part II Vorticity 12.1 The vorticity equation 12.2 The Biot-Savart Law 12.3 The Beale-Kato-Majda blowup criterion 12.4 The vorticity in two dimensions 12.5 A local version of the Biot-Savart Law Notes Exercises The Serrin condition for local regularity 13.1 Local weak solutions 13.2 Main auxiliary theorem: a smallness condition 13.3 The case ^ + - 1 13.4 The case ^ + - = 1 4 î 13.5 Local Holder regularity in time for spatially smooth u Notes Exercise The local energy inequality 14.1 Formal derivation of the local energy inequality 14.2 The Leray régularisation 14.3 Rigorous derivation of the local energy inequality 14.4 Derivation of an alternative local energy inequality 14.5 Derivation of the strong energy inequality on R 3 Notes Exercises Partial regularity I: dimB(S) 5/3 15.1 Scale-invariant quantities 15.2 Outline of the proof 15.3 A first local regularity theorem in terms of u and p 15.4 Partial regularity I: dimg (S) 5/3 15.5 Lemmas for the first partial regularity theorem Notes Exercises
Contents xi 16 Partial regularity II: dimnC?) 1 315 16.1 Outline of the proof 316 16.2 A second local regularity theorem 321 16.3 Partial regularity II: H l ( S ) = 0 326 16.4 The Serrin condition revisited: u e L t °°Z^ 329 16.5 Lemmas for the second partial regularity theorem 333 Notes 338 Exercises 338 17 Lagrangian trajectories 340 17.1 Lagrangian trajectories for classical solutions 342 17.2 Lagrangian uniqueness for ito € H fl H x/1 343 17.3 Existence of a Lagrangian flow map for weak solutions 352 17.4 Lagrangian a.e. uniqueness for suitable weak solutions 358 17.5 Proof of the inequality (17.5) 363 17.6 Proof of the borderline Sobolev embedding inequality 366 Notes 367 Exercises 368 Appendix A Functional analysis: miscellaneous results 369 A. 1 U spaces 369 A.2 Absolute continuity 370 A.3 Convolution and mollification 371 A.4 Weak U spaces 373 A.5 Weak and weak-* convergence and compactness 374 A. 6 Gronwall’s Lemma 377 Appendix B Calderon-Zygmund Theory 378 B. l Calderdn-Zygmund decompositions 378 B.2 The Calderdn-Zygmund Theorem 380 B. 3 Riesz transforms 384 Appendix C Elliptic equations 387 C. l Harmonic and weakly harmonic functions 387 C. 2 The Laplacian 388 Appendix D Estimates for the heat equation 393 D. 1 Existence, uniqueness, and regularity 393 D.2 Estimates for e A o 394 D.3 Estimates for (3, — A) -1 ƒ 396
Contents xii D.4 Higher regularity - Holder estimates 400 D.5 Maximal regularity for the heat equation 403 Appendix E A measurable-selection theorem 407 Solutions to exercises 412 References 457 Index 467
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| any_adam_object | 1 |
| author | Robinson, James C. 1969- Rodrigo, José L. Sadowski, Witold 1973- |
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| author_facet | Robinson, James C. 1969- Rodrigo, José L. Sadowski, Witold 1973- |
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| author_sort | Robinson, James C. 1969- |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV043815378 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-20T17:45:54Z |
| institution | BVB |
| isbn | 9781107019669 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-029226512 |
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| physical | xiv, 471 Seiten Diagramme |
| publishDate | 2016 |
| publishDateSearch | 2016 |
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| publisher | Cambridge University Press |
| record_format | marc |
| series | Cambridge studies in advanced mathematics |
| series2 | Cambridge studies in advanced mathematics |
| spellingShingle | Robinson, James C. 1969- Rodrigo, José L. Sadowski, Witold 1973- The three-dimensional Navier-Stokes equations classical theory Cambridge studies in advanced mathematics Navier-Stokes-Gleichung (DE-588)4041456-5 gnd Dreidimensionale Strömung (DE-588)4150635-2 gnd |
| subject_GND | (DE-588)4041456-5 (DE-588)4150635-2 |
| title | The three-dimensional Navier-Stokes equations classical theory |
| title_auth | The three-dimensional Navier-Stokes equations classical theory |
| title_exact_search | The three-dimensional Navier-Stokes equations classical theory |
| title_full | The three-dimensional Navier-Stokes equations classical theory James C. Robinson (University of Warwick), José L. Rodrigo (University of Warwick), Witold Sadowski (University of Warsaw) |
| title_fullStr | The three-dimensional Navier-Stokes equations classical theory James C. Robinson (University of Warwick), José L. Rodrigo (University of Warwick), Witold Sadowski (University of Warsaw) |
| title_full_unstemmed | The three-dimensional Navier-Stokes equations classical theory James C. Robinson (University of Warwick), José L. Rodrigo (University of Warwick), Witold Sadowski (University of Warsaw) |
| title_short | The three-dimensional Navier-Stokes equations |
| title_sort | the three dimensional navier stokes equations classical theory |
| title_sub | classical theory |
| topic | Navier-Stokes-Gleichung (DE-588)4041456-5 gnd Dreidimensionale Strömung (DE-588)4150635-2 gnd |
| topic_facet | Navier-Stokes-Gleichung Dreidimensionale Strömung |
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